A variety of compact sorting machines exist for receiving and sorting sheets from am output slot of a paper sheet processing machine, such as a copy machine. Generally, these sorting machines provide a number of trays supported in a compact vertical stack, each of which, in turn, is capable of traversing past the sheet output slot. To accommodate paper entry into each receiving tray in turn, a relatively large gap is required between the receiving tray and the tray immediately above the receiving tray. Tray spacing elsewhere may be minimized in order to increase the compactness of the vertical stack. Several tray shifting methods have been devised to accomplish this variable tray spacing requirement.